Last night I was introduced through Slashdot to Lockhart’s Lament, an essay written by a disgruntled mathematics educator on how far what is taught in math classes differs from the practice of mathematics. Although a bit daunting at 25 pages, it is well worth the time. I do not agree with Lockhart that his particular field of expertise is more poorly covered than others in primary and secondary schools, but he provides a compelling example of one. I am also unconvinced that the heart of mathematics is quite as artistic and unrelated to practicality as he would have you believe; he easily admits that to do mathematics is to discover or invent ways to solve problems, and I believe that throughout history those problems have been largely inspired by practical concerns. Finally, I disagree with Lockhart about the importance of learning to perform arithmetic at a young age. Perhaps we could agree that this should be called something other than mathematics, but I believe it is nearly as important as the ability to read, write, and speak natural language.
I find myself wholly in agreement with Lockhart, however, on his main premise: that mathematics is not and should not be the memorization of facts and theorems or turning oneself into an automaton capable of performing calculations by following repetition of an algorithm without understanding it. To ask a question and to discover an answer is where real thinking is learned. Even when the student fails to find an answer, to have grappled with the problem and understood it, then given a well-known answer devised by some long-dead Greek with an explanation of the thought process that led to it is orders of magnitude more enlightening than to simply be told the results of that work. The whole point of recording discoveries is so that future generations will not need to spend their lifetimes working on the same problem, but not to forget the reasoning behind the solution.
I recall that in high school one of my friends came to find out that some of the strangest constants we had learned about were related in a highly unusual manner by what I now know to be Euler’s identity. My recollection is that he found this to be the case by randomly playing with these numbers in his calculator while the teacher was covering something far less interesting, but that may be incorrect. In any case, we asked our mathematics teacher about this and she was unable to give a satisfactory response. If my memory is correct we determined that it had something to do with polar coordinates and believed a full explanation to be beyond our current understanding. I had forgotten about it since then and, to this day, have no real intuition as to why it might be true; it is simply a fact that I know.
In another mathematical anecdote, at some point in the last year I was attempting to solve some practical problem in combinatorics. In the process of laborious calculations to solve this problem, I unintentionally derived the identity that the sum of all positive integers less than or equal to n is n(n+1)/2. At first I was frustrated about the time I could have saved if I had remembered this fact, but then I found myself quite impressed with the utility and simplicity of the identity itself; I rather doubt that I will forget it again. If it had been taught to me in this way, I doubt I would have done so even once.
Even before reading Mr. Lockhart’s essay I harbored feelings of this sort about science education. To do science is not simply to know things, but to find ways to determine things that are unknown. In my science courses in high school and even the two physics courses I took in college we spent plenty of time in the lab performing experiments and observing results. In exceptional cases we may have even hypothesized as to what results we expected and explained why, but none of this was really science. Performing experiments is the grunt work of scientific discovery; formulating reasonable and interesting hypotheses, and designing the experiments to prove or disprove them, is actual science. I do not recall doing much, if any, of that in school
Actually, the things we did as fun exercises, not thinking of them as teaching us anything (certainly nothing on which we could be tested) were much closer to awakening a scientific mind. Trying to build the strongest structure possible out of marshmallows and toothpicks does not exactly follow the scientific method, but it requires critical thinking about the directions and magnitudes of forces and strengths of materials, and observing the outcomes can open up all sorts of questions about why one’s intuitions did not match reality.
Some disciplines seem to do a better job of this than others. English classes certainly involved students actively writing, reading, and critiquing literature, for example. Thankfully, I think that computer scientists actually do a better job of this than most. Even in our most introductory courses we essentially give students a few basic building blocks and have them go solve problems in their own way, hopefully giving them advice on how to construct more elegant or efficient solutions, but not training them to imitate the masters without any real understanding of what they are doing.